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Stream: learning: questions

Topic: Categorification of eigenvalues and eigenvectors

fosco (Apr 20 2025 at 15:30):

I want to submit to you the following shenanigan which I find very funny to play with.

Let $p : Y^o\times Y\to Set$ be an endoprofunctor on a category $Y$. Let $\Lambda$ a set, and denote still as $\Lambda$ the costant functor at $\Lambda$.

• $\Lambda$ is an eigenvalue of $p$ if the category $(p/\Lambda)$ of natural transformations $p \Rightarrow \Lambda$ is nonempty and... something else I'd like to figure out so that what follows makes sense (and avoids vacuous truths of sort).
• A functor $F : Y^{op}\to Set$ is an eigenvector wrt $\Lambda$ if, when $\Lambda$ is an eigenvalue, for every $p\Rightarrow\Lambda$ in $(p/\Lambda)$, there is an initial cowedge $p(x,y)\times Fy \to \Lambda\times Fx$ (so that the "matrix product" $(p\otimes F)x:= \int^y p(x,y)\times Fy$ between the matrix $p$ and the vector $F$ is canonically isomorphic to the product $\Lambda\times F$ of the "scalar" $\Lambda$ and of $F$, evaluated at $x$, as prescribed by the theory of eigenvalues/eigenvectors).

fosco (Apr 20 2025 at 15:33):

I have a hunch that connectedness, or some sort of contractibility, of $(p/\Lambda)$ could be required; I also suspect the second property is too strong, but note that (for example) with this definition the eigenspace of $\Lambda$ for $p$ (the subcategory spanned by all eigenvectors wrt $\Lambda$) is closed under colimits, so it can't be too trivial a subcategory of the small-cocomplete category $[Y^{op},Set]$. Note also that the only eigenvector of hom is the terminal object, with eigenspace the whole category, as god intended.

fosco (Apr 20 2025 at 15:35):

I remember a few people that, like me, love categorifying linear algebra; @Simon Willerton might have tinkered with this idea, or maybe @Jade Master (who I remember was blogging about the analogies between matrices and quantale-enriched profunctors?)

Matteo Capucci (he/him) (Apr 30 2025 at 10:27):

Do you have examples to aid intuition on what these look like?

fosco (Apr 30 2025 at 12:00):

Well, if you squint your eyes a profunctor is a matrix.. and the coend computation yields almost instantaneosuly that if a profunctor results as $p=L\times R$ for a copresheaf $L$ and a presheaf $R$, then $p$ has eigenvalue the functor tensor product $L\otimes R=\int^x Lx\times Rx$, with associated eigenvector $L$.

Compare with the analogue fact of life, that if a matrix $A$ results as the tensor product $v\otimes w$, then $A$ has the dot product $v\cdot w$ as eigenvalue, and eigenspace $\langle v\rangle$ (the other eigenvalue is 0, with eigenspace an orthogonal complement of v... that's more difficult to categorify).

fosco (Apr 30 2025 at 12:01):

I have a list, now more abundant than days ago, of similar analogies. So, this problem ended up being so pretty it can be its own motivation -and I see only one path to aid intuition: trust the coends you write, and see what comes out of them.

Matteo Capucci (he/him) (Apr 30 2025 at 17:55):

I wonder if working with spans of $\infty\rm Grpd$ as in https://arxiv.org/abs/1602.05082 might make the analogies smoother

David Corfield (May 01 2025 at 10:06):

Probably a distraction, but since @Matteo Capucci (he/him) brought up that paper, 'Homotopical Linear Algebra', something I meant to return to was a couple of long comments made by Joachim Kock on the $n$-Category Café, here, which passed seemingly unobserved.

Wherever I read Joachim, he seems to be pushing things in a homotopical direction. E.g., there's talk of some unavoidable 'homotopy overhead' in Whole-grain Petri nets and processes:
image.png

Then back here, he jumps in quickly to support a hunch I'd had on some result Tom Leinster was airing on commuting limits and colimits over groups.